When the temperature or stress around the FBG changes, it will cause changes to the grating pitch period and the refractive index of the fiber core. In this case, the center wavelength of the fiber Bragg grating will move. According to equation (1), the FBG center wavelength shift can be expressed as:

$$\begin{array}{rl}\mathrm{\Delta}{\lambda}_{B}& =2{n}_{eff}\mathrm{\Delta}\mathrm{\Lambda}+2\mathrm{\Lambda}\mathrm{\Delta}{n}_{eff}\\ & =2{n}_{eff}\mathrm{\Lambda}\frac{\mathrm{\Delta}\mathrm{\Lambda}}{\mathrm{\Lambda}}+2\mathrm{\Lambda}{n}_{eff}\frac{\mathrm{\Delta}{n}_{eff}}{{n}_{eff}}={\lambda}_{B}\frac{\mathrm{\Delta}\mathrm{\Lambda}}{\mathrm{\Lambda}}+{\lambda}_{B}\frac{\mathrm{\Delta}{n}_{eff}}{{n}_{eff}}\end{array}$$(2)Assumptions: The axial stress field of the FBG remains stable, and the reflected light center wavelength is only affected by the temperature field. Differentiating with respect to temperature on both sides of equation (2) yields the corresponding relationship between reflection wavelength and temperature:

$$\begin{array}{rl}\frac{d{\lambda}_{B}}{dT}& ={\lambda}_{B}\frac{1}{\mathrm{\Lambda}}\frac{d\mathrm{\Lambda}}{dT}+{\lambda}_{B}\frac{1}{{n}_{eff}}\frac{d{n}_{eff}}{dT}\\ & =(\alpha +\zeta )\times {\lambda}_{B}\end{array}$$(3)In equation 3, *𝛼* is thermal expansion coefficient of optical fiber material, $\alpha ={\displaystyle \frac{1}{\mathrm{\Lambda}}}{\displaystyle \frac{d\mathrm{\Lambda}}{dT}}$

*𝜁* is thermal optical coefficient of optical fiber material, $\zeta ={\displaystyle \frac{1}{{n}_{eff}}}{\displaystyle \frac{d{n}_{eff}}{dT}}$.

When the temperature changes, the influence of the thermo-optic effect and the thermal expansion on the effective refractive index of the grating period and mode is mainly due to change of the material scale, and the grating period. According to formula (3), the temperature sensitivity coefficient of FBG can be expressed as:

$$\begin{array}{r}{K}_{T}=\frac{1}{{\lambda}_{B}}\frac{d{\lambda}_{B}}{dT}=\alpha +\zeta \end{array}$$(4)The FBG temperature sensitivity reflects the relationship between wavelength relative drift *𝛥*_{B}/*𝜆*_{B} and *𝛥T*. The thermal expansion coefficient and thermo-optic coefficient of FGB are constant at a certain temperature range, and the wavelength change of the reflected FBG is linear with the temperature change. When the material is determined, *K*_{T} is a constant associated with the material. When at room temperature, the effective refractive index of the FBG of the quartz core is *n*_{eff} = 1.4469, the thermal expansion coefficient *𝛼* is 0.5 × 10^{−6}/^{∘}*C*, and the thermal coefficient is *𝜁* = 8.3 × 10^{−6}/^{∘}*C* to 9.5 × 10^{−6}/^{∘}*C*. The theoretical value of the temperature sensitivity coefficient *K*_{T} is approximately 8.8 × 10^{−6}/^{∘}*C* to 10 × 10^{−6}/^{∘}*C*, while the temperature sensitivity coefficient *K*_{T} of the silicon optical fiber is approximately 6.67 × 10^{−6}/^{∘}*C*.

The relationship between the center wavelength of the FBG and the temperature is:

$$\frac{\mathrm{\Delta}{\lambda}_{B}}{{\lambda}_{B}}=(\alpha +\zeta )\times \mathrm{\Delta}T$$(5)Therefore, when under the condition that the axial stress field remains stable, there is a good linear relationship between the reflection wavelength and temperature of the FGB sensor. By detecting the change of the center wavelength of the grating, the temperature value of the measured point can be calculated.

## Comments (0)

General note:By using the comment function on degruyter.com you agree to our Privacy Statement. A respectful treatment of one another is important to us. Therefore we would like to draw your attention to our House Rules.